Question

Analytic signal, \(A(\theta)\), for \(f(\theta) = \cos\theta\) is equal to

• A. \(A(\theta) = -\cos\theta\)
• B. \(A(\theta) = -\sin\theta\)
• C. \(A(\theta) = e^{i\theta}\)
• D. \(A(\theta) = -e^{i\theta}\)

Hint:

The analytic signal is \(f(\theta)+i\hat f(\theta)\); the Hilbert transform of \(\cos\theta\) is \(\sin\theta\), giving Euler's identity.

Answer 1

The Correct Option is C

The analytic signal of a real function \(f(\theta)\) is built by adding \(i\) times its Hilbert transform to itself: \(A(\theta) = f(\theta) + i\,\hat f(\theta)\), where \(\hat f\) denotes the Hilbert transform of \(f\).

Step 1: Hilbert transform of \(\cos\theta\).
The Hilbert transform acts as a 90-degree phase-delay filter: it turns \(\cos\theta\) into \(\sin\theta\) (and, consistently, turns \(\sin\theta\) into \(-\cos\theta\), so that applying it twice gives back \(-f(\theta)\), as required of any Hilbert-transform pair). So \(\hat f(\theta) = \sin\theta\).

Step 2: Build the analytic signal.
\(A(\theta) = \cos\theta + i\sin\theta\).

Step 3: Recognize the complex exponential.
By Euler's formula, \(\cos\theta + i\sin\theta = e^{i\theta}\).

So \(A(\theta) = \boxed{e^{i\theta}}\) - option (C).